Let $A$ and $B$ are square matrices of order $2$ such that $A+adj\left(B^T\right)=\left[\begin{array}{ll}3& 2\\ 2& 3\end{array}\right]$ and $A^T-adj\left(B\right)=\left[\begin{array}{ll}-2& -1\\ -1& -2\end{array}\right]$, then $A^2+2A^3+3A^4+5A^5$ is equal to (where $M^T$ and $adj\left(M\right)$ represent the transpose matrix and adjoint matrix of matrix $M$ respectively and $I$ represents the identity matrix of order $2$)